Cost gap, Shapley, or nucleolus allocation: Which is the best game‐theoretic remedy for the low‐risk anomaly?

Abstract

Empirical research in investment management has discovered the puzzling phenomenon that, contrary to established capital market theory, low-risk assets tend to earn larger returns than their high-risk counterparts. In a recent contribution, Auer and Hiller (2019) emphasize that an inadequate quantification of risk may be the root of this problem. By interpreting a portfolio as a cooperative game, they arrive at the interesting finding that using assets' risk-based Shapley values instead of classic stand-alone risk measures has the potential to solve the low-risk puzzle. In this article, we extend their study first by considering additional game-theoretic risk allocation schemes, namely, the cost gap technique and the nucleolus method, and then by improving their simulation design via a larger number of assets and a supplementary determination of risk-return slopes. We find that the Shapley method outperforms the alternatives with respect to the generation of positive risk-return slopes. JEL CLASSIFICATIONG10; G11; C71 INTRODUCTIONFinancial research has revealed many capital market phenomena, which challenge the validity of the traditional asset pricing theory

Portfolio structures: Can they contribute to solving the low‐risk puzzle?

“…2 1] for the assets) to the risk allocation vector (computed as described above). In a very similar way, Auer and Hiller (2021)…”

Allocation of portfolio risk and outside options

“…The Shapley value has been applied to the problem of risk allocation in papers by Mussard and Terraza (2008), Ortmann (2016), Balog et al (2017), Auer and Hiller (2019, 2021), and Shalit (2020).…”

“…These include the activity based method (Hamlen et al, 1977), the Beta method (Homburg & Scherpereel, 2008, for example) and the incremental approach (see, for instance, Jorion, 1985). In addition to these concepts, a growing body of literature has emerged that uses cooperative game theory, especially the Shapley value (Shapley, 1953), the nucleolus (Schmeidler, 1969), and the τ value — also known as cost gap method (Tijs & Driessen, 1986; Tijs, 1987), for risk allocation (Auer & Hiller, 2019; 2021; Balog et al, 2017; Mussard & Terraza, 2008; Ortmann, 2016; 2018). All approaches distribute portfolio risk to individual assets under the assumption that the portfolio is the entire stock market respectively the surrounding stock market has no influence on the distribution of portfolio risk.…”

“…One reason for this phenomenon may be the difficulty in allocating portfolio risk to individual assets. Auer and Hiller (2019, 2021) showed in simulation studies that using the Shapley value instead of classic measures of risk has the potential to solve the low‐risk puzzle.…”

“…For our analysis, we perform a simulation study based on the ideas of Auer and Hiller (2019, 2021). We look at an investor who is interested in combining individual assets in a portfolio and evaluate the riskiness of each asset based on its χ payoff with respect to the portfolio variance.…”